It's very amusing to consider instead the function
[tex]
e^{-t/5}
[/tex]
It looks almost just like your function but with a few less little "fine structure" wiggles.
The integrals of both functions should be rather similar. For example
[tex]
\int_0^{\infty} e^{(-t/5)}=5
[/tex]
whereas
[tex]
\int_0^{\infty}e^{(Cos[t]-t)/5}\approx 5.09
[/tex]